CMR : A = 1/3 +2/32 + 3/33 + 4/34 + 5/35 +...+102/3102<3/4
chứng minh rằng A=1+3+31+32+33+34+.....+3102+3103chia hết cho 4
\(A=1+3+3^2+3^3+...+3^{102}+3^{103}\)
\(\Rightarrow A=\left(1+3\right)+\left(3^2+3^3\right)+...+\left(3^{102}+3^{103}\right)\)
\(\Rightarrow A=\left(1+3\right)+3^2\left(1+3\right)+...+3^{102}\left(1+3\right)\)
\(\Rightarrow A=\left(1+3\right)\left(1+3^2+...+3^{102}\right)\)
\(\Rightarrow A=4\left(1+3^2+...+3^{102}\right)⋮4\)
Chứng minh rằng:
A = 1/3 + 1/32 + 1/33 + ..........+ 1/399 < 1/2
B = 3/12x 22 + 5/22 x 32 + 7/32 x 42 +............+ 19/92 x 102 < 1
C = 1/3 + 2/32 + 3/33 + 4/34 +.........+ 100/3100 ≤ 0
\(A=\dfrac{1}{3}+\dfrac{1}{3^2}+\dfrac{1}{3^3}+\dfrac{1}{3^4}+...+\dfrac{1}{3^{99}}\)
\(\Rightarrow\dfrac{A}{3}=\dfrac{1}{3^2}+\dfrac{1}{3^3}+\dfrac{1}{3^4}+...+\dfrac{1}{3^{100}}\)
\(\Rightarrow A-\dfrac{A}{3}=\dfrac{2A}{3}=\left(\dfrac{1}{3}+\dfrac{1}{3^2}+\dfrac{1}{3^3}+...+\dfrac{1}{3^{99}}\right)-\left(\dfrac{1}{3^2}+\dfrac{1}{3^3}+\dfrac{1}{3^4}+...+\dfrac{1}{3^{100}}\right)\)
\(\Rightarrow\dfrac{2A}{3}=\left(\dfrac{1}{3^2}-\dfrac{1}{3^2}\right)+\left(\dfrac{1}{3^3}-\dfrac{1}{3^3}\right)+...+\left(\dfrac{1}{3^{99}}-\dfrac{1}{3^{99}}\right)+\left(\dfrac{1}{3}-\dfrac{1}{3^{100}}\right)=\dfrac{1}{3}-\dfrac{1}{3^{100}}\)
\(\Rightarrow2A=3\cdot\left(\dfrac{1}{3}-\dfrac{1}{3^{100}}\right)\)
\(\Rightarrow\text{A}=\dfrac{1-\dfrac{1}{3^{99}}}{2}\)
\(\Rightarrow A=\dfrac{1}{2}-\dfrac{1}{2.3^{99}}< \dfrac{1}{2}\)
Cho \(A=1+3+3^2+3^3+3^4+...+3^{90}\) CMR \(A\) không phải là số chính phương
Lời giải:
$A=1+3+3^2+(3^3+3^4+3^5+3^6)+(3^7+3^8+3^9+3^{10})+...+(3^{87}+3^{88}+3^{89}+3^{90})$
$=13+3^3(1+3+3^2+3^3)+3^7(1+3+3^2+3^3)+....+3^{87}(1+3+3^2+3^3)$
$=13+(1+3+3^2+3^3)(3^3+3^7+...+3^{87})$
$=13+40(3^3+3^7+...+3^{87})$
$\Rightarrow A$ chia 5 dư 3
Do đó A không là scp.
Ta có:
\(A=1+3+3^2+3^3+...+3^{90}\)
\(3A=3\cdot\left(1+3+3^2+...+3^{90}\right)\)
\(3A=3+3^2+3^3+...+3^{91}\)
\(3A-A=3+3^2+3^3+...+3^{91}-1-3-3^2-...-3^{90}\)
\(2A=3^{91}-1\)
\(A=\dfrac{3^{91}-1}{2}\)
Mà: \(3^{91}-1\) không phải là số chính phương nên \(A=\dfrac{3^{91}-1}{2}\) không phải là số chính phương
Bài 5: Thực hiện phép tính:
a) 47 – [(45.24 – 52.12):14] | a) 3.52 + 15.22 – 26:2 |
b) 50 – [(20 – 23) : 2 + 34] | b) (- 25) . (- 4 ) + (- 70) . 3 |
c) 102 – [60 : (56 : 54 – 3.5)] | c) 62 : (- 9) - 50.2 – 33.(- 3) |
d) 50 – [(50 – 23.5):2 + 3] | d) 32.(- 50) - 23.10 – 81: (- 3) |
e) 10 – [(82 – 48).5 + (23.10 + 8)] : 28 | e) 513 : 510 – 25.22 |
f) 8697 – [37 : 35 + 2(13 – 3)] | f) 20 : 22 + 59 : 58 |
g) 2011 + 5[300 – (17 – 7)2] | g) 100 : 52 + 7.32 |
h) 695 – [200 + (11 – 1)2] | h) 84 : 4 + 39 : 37 + 50 |
i) 129 – 5[29 – (6 – 1)2] | i) 29 – [16 + 3.(51 – 49)] |
j) 2010 – 2000 : [486 – 2(72 – 6)] | j) 53.2 – 100 : 4 + 23.5 |
f: =20:4+5
=5+5
=10
e: =125-100=25
Giúp mình giải nhoa :*
Tính nhanh
a)1-3+5-7+9-11+....33-35.
B) (42+43+44+45++47)-(32+33+34+35+36+37)
a,1-3+5-7+9-.......+33-35
=(1+5+9+....+33)-(3+7+11+...+35)
=153-171
=-18
Tick mk vài cái lên 300 mk giải nốt phần b
a) Đặt A=1-3+5-7+9-11+.......+33-35
Ta có: A=(1-3)+(5-7)+.........+(33-35)
=> A=(-2)+(-2)+.........+(-2)
18 số hạng
=> A=(-2).18
=> A=-36
b) Đặt B= (42+43+44+45+36+47)-(32+33+34+35+36+37)
Ta có: B=(42-32)+(43-33)+(44-34)+(45-35)+(47-37)+(46-36)
=> B= 10+10+10+10+10+10
=> B=10.6
=> B=60
1+32+34+36+...+3100+3102
\(A=1+3^2+3^4+...+3^{102}\)
\(9A=3^2+3^4+...+3^{102}+3^{104}\)
\(\Rightarrow9A-A=3^{104}-1\)
\(\Rightarrow8A=3^{104}-1\)
\(\Rightarrow A=\dfrac{3^{104}-1}{8}\)
Cho S = 1 + 3 + 32 + 33 + 34 + 35 + 36 + 37 + 38 + 39. Chứng tỏ rằng S chia hết cho 4.
\(S=\left(1+3\right)+...+3^8\left(1+3\right)=4\left(1+...+3^8\right)⋮4\)
Cho S = 1 + 3 + 32 + 33 + 34 + 35 + 36 + 37 + 38 + 39. Chứng tỏ rằng S chia hết cho 4.
\(S=1+3+3^2+3^3+3^4+3^5+3^6+3^7+3^8+3^9\)
\(S=\left(1+3\right)+\left(3^2+3^3\right)+\left(3^4+3^5\right)+\left(3^6+3^7\right)+\left(3^8+3^9\right)\)
\(S=4+3^2\left(1+3\right)+3^4\left(1+3\right)+3^6\left(1+3\right)+3^8\left(1+3\right)\)
\(S=4+3^2.4+3^4.4+3^6.4+3^8.4\)
\(S=4\left(3^2+3^4+3^6+3^8\right)\)
\(4⋮4\\ \Rightarrow4\left(3^2+3^4+3^6+3^8\right)⋮4\\ \Rightarrow S⋮4\)
1+2+3+4+5+6+7+8+9+10+11+12+13+14+15+16+17+18+19+20+21+22+23+24+25+26+27+28+29+30+31+32+33+34+35+36+37=
1+2+3+4+5+6+7+8+9+10+11+12+13+14+15+16+17+18+19+20+21+22+23+24+25+26+27+28+29+30+31+32+33+34+35+36+37
=(1+37)x37:2
=703
Điền vào ô vuông các dấu thích hợp (=; <; >):
a) 2 3 . 5 + 3 4 . 2 - 4 . ( 5 7 : 5 5 ) □ 15 : ( 3 5 : 3 4 ) + 5 . 2 4 - 7 2 - 4 ;
b) ( 3 5 . 3 7 ) : 3 1 0 + 5 . 2 4 □ 5 . 2 2 . 2 3 - 4 . ( 5 8 : 5 6 ) ;
c) 2 [ ( 7 - 3 3 : 3 2 ) : 2 2 + 99 ] - 100 □ 3 4 . 2 + 2 3 . 5 - 7 ( 5 2 - 5 ) ;
d) 207 : { 2 ^ 3 . [ ( 156 - 128 ) : 14 ] + 7 ] □ 117 : { [ 79 - 3 ( 3 ^ 3 - 17 ) ] : 7 + 2 }